# How is the speed of a chemical reaction related to the spontaneity of the reaction?

Aug 22, 2016

It's not. Spontaneity is a thermodynamic phenomenon, and the rate/speed of a reaction is a kinetic phenomenon.

The rate of a chemical reaction can be written in a rate law as

$r \left(t\right) = k \left[A\right]$

where

• $r \left(t\right)$ is the rate of reaction.
• $k$ is the rate constant for that reaction at a given temperature.
• $\left[A\right]$ is the concentration of $A$ in $\text{M}$.

for a simple unimolecular reaction

$A \to B$,

which has activation energy ${E}_{a}$. Reactions also have a Gibbs' free energy of reaction, $\Delta {G}_{\text{rxn}}$.

Both are on the same reaction coordinate diagram, but they do not influence each other. Modifying the rate $r \left(t\right)$ modifies the rate constant $k$ when $\left[A\right]$ is held constant (such as changing the temperature, pressure, etc).

From the Arrhenius equation

$k = A {e}^{- {E}_{a} \text{/RT}}$,

increasing $k$ increases ${e}^{- {E}_{a} \text{/RT}}$, which means ${E}_{a}$ is decreased. Hence, a lower activation energy indicates a faster reaction.

Yes, $\Delta {G}_{\text{rxn}}$ indicates a reaction is spontaneous when $\Delta {G}_{\text{rxn}} < 0$, but it doesn't matter.

When you modify ${E}_{a}$, you do not modify the free energy of the reactants or products, so you do not touch $\Delta {G}_{\text{rxn}}$ for the overall reaction. If you raise/lower the peak of the hill, you have not raised/lowered the bottom of the hill.

• If the overall reaction was spontaneous without a catalyst, it is still spontaneous with a catalyst.
• If it is also slow without a catalyst, it is now fast with a catalyst, but not additionally and suddenly possible/spontaneous.